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On Complex and Noncommutative Tori PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 4, Pages 973–981 S 0002-9939(05)08244-4 Article electronically published on September 28, 2005 ON COMPLEX AND NONCOMMUTATIVE TORI IGOR NIKOLAEV (Communicated by Michael Stillman) Abstract. The “noncommutative geometry” of complex algebraic curves is studied. As a first step, we clarify a morphism between elliptic curves, or ∗ 2πiθ complex tori, and C -algebras Tθ = {u, v | vu = e uv},ornoncommutative tori. The main result says that under the morphism, isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra of Riemann surfaces. Introduction Noncommutative geometry is a branch of algebraic geometry studying “varieties” over noncommutative rings. The noncommutative rings are usually taken to be rings of operators acting on a Hilbert space [7]. The rudiments of noncommutative geometry can be traced back to F. Klein [3], [4] or even earlier. The fundamental modern treatise [1] gives an account of status and perspective of the subject. ∗ The noncommutative torus Tθ is a C -algebra generated by linear operators u and v on the Hilbert space L2(S1) subject to the commutation relation vu = e2πiθuv, θ ∈ R − Q [11]. The classification of noncommutative tori was given in [2], [8], [11]. Recall that two such tori Tθ,Tθ are Morita equivalent if and only if θ, θ lie in the same orbit of the action of group GL(2, Z) on irrational numbers by linear fractional transformations. It is remarkable that the “moduli problem” for Tθ looks as such for the complex tori Eτ = C/(Z + τZ), where τ is complex modulus. Namely, complex tori Eτ ,Eτ are isomorphic if and only if τ,τ lie in the same orbit of the action of SL(2, Z) on complex numbers by linear fractional transformations. It was observed by some authors (e.g. [5], [15]) that it might not be just a coincidence. This note is an attempt to show that it is indeed so: there exists a general morphism between Riemann surfaces and C∗-algebras. Let us give rough idea of our approach. Given Riemann surface S,thereisa → R∞ function S + which maps the (discrete) set of closed geodesics of S to a discrete subset of a real line by assigning each closed geodesic its riemannian length. If Tg(S) is the space of all Riemann surfaces of genus g ≥ 0, then the function −→ R∞ (1) W : Tg(S) + Received by the editors February 25, 2003 and, in revised form, November 2, 2004. 2000 Mathematics Subject Classification. Primary 14H52, 46L85. Key words and phrases. Elliptic curve, noncommutative torus. c 2005 American Mathematical Society 973 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 974 IGOR NIKOLAEV is finite-to-one and “generically” one-to-one [17]. In the case g = 1, function W is always one-to-one. It is known also that restriction Wsyst : Tg(S) → R+ of W to 0 the shortest closed geodesic of S (called systole)isaC Morse function on Tg(S) [13], §5. Below we focus on the case g = 1, i.e. T1 Eτ . Recall that Tθ has a unique state s0 (which is actually a tracial state) [11]. Any positive functional on Tθ has form ωs0,whereω>0 is a real number. Let Θ={Tθ | θ ∈ R − Q} and Ω = {ω ∈ R | ω>0}. We define a map × −→ R∞ (2) V :Θ Ω + by the formula (T ,ω) →{f (ω)lntr(A )}∞ ,where θ n n n=0 a0 1 A0 = , 10 a0 1 a1 1 A1 = 10, 10, . (3) . a0 1 a1 1 an 1 An = 10 10... 10 are integer matrices whose entries ai > 0 are partial denominators of the continued 0 fraction expansion of θ,andfn are monotone C functions of ω. Assuming that −1 functions W, V have common range, one gets a mapping WV : Eτ → (Tθ,ω). Morphisms between Eτ and Tθ have been studied in [5], [9], [10], [16]. The works [5], [10] and [16] treat noncommutative tori as “quantum compactification” of the space of elliptic curves. This approach deals with an algebraic side of the subject. In particular, Manin [5] suggested to use “pseudolattices” (i.e. K0-group of Tθ)to solve the multiplication problem for real number fields. This problem is part of the Hilbert 12th problem. In [9] a functor from derived category of holomorphic vector bundles over Tθ to the Fukaya category of such bundles over Eτ was constructed. In this note we prove the following results. Theorem 1. Let Eτ be a complex torus of modulus τ,Im τ > 0,andlet(Tθ,ω) be a pair consisting of noncommutative torus with an irrational Rieffel’s parameter θ and a positive functional Tθ → C of norm ω. Then there exists a one-to-one mapping Eτ → (Tθ,ω). The action of the modular group SL(2, Z) on the complex half plane {τ ∈ C | Im τ > 0} is equivariant with: (i) the action of group GL(2, Z) on irrationals {θ ∈ R − Q | θ>0} by linear fractional transformations; (ii) a discrete action on positive reals {ω ∈ R | ω>0}. In particular, isomorphic complex tori map to the Morita equivalent noncommuta- tive tori, and vice versa. Definition 1. The irrational number θ of mapping Eτ → (Tθ,ω) we call a projec- tive curvature of the elliptic curve Eτ . Theorem 2. Projective curvature of an elliptic curve with complex multiplication is a quadratic irrationality. 1. Proofs The proof of both theorems is based on the rigidity of length spectrum of complex torus; cf. Wolpert [17]. Preliminary information on complex and noncommutative tori can be found in Section 2. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON COMPLEX AND NONCOMMUTATIVE TORI 975 1.1. Proof of Theorem 1. Let us review the main steps of the proof. By the rigidity lemma (Lemma 1) the length spectrum Sp E defines conformal structure of E. In fact, this correspondence is a bijection. Under isomorphisms of E the length spectrum can acquire a real multiple or get a “cut of finite tail” (Lemma 2). We attach to C/L a continued fraction of its projective curvature θ as specified in the Introduction. Then isomorphic tori C/L will have continued fractions which differ only in a finite number of terms. In other words, one can attach a Morita equivalence class of noncommutative tori to every isomorphism class of complex tori. Lemma 1 (Rigidity of length spectrum). Let Sp E be the length spectrum of a complex torus E = C/L. Then there exists a unique complex torus with the spectrum. This correspondence is a bijection. Proof. See McKean [6]. Let Sp X = {l1,l2,...} be the length spectrum of a Riemann surface X.Leta> 0bearealnumber.ByaSp X we understand the length spectrum {al1,al2,...}. Similarly, for any m ∈ N we denote by Spm X the length spectrum {lm,lm+1,...}, i.e. the one obtained by deleting the first (m − 1)-geodesics in Sp X. Lemma 2. Let E ∼ E be isomorphic complex tori. Then either: (i) Sp E = |α|Sp E for an α ∈ C×,or (ii) Sp E = Spm E for a m ∈ N. Proof. (i) The complex tori E = C/L, E = C/M are isomorphic if and only if M = αL for a complex number α ∈ C×. It is not hard to see that the closed geodesic of E are bijective with the points of the lattice L = ω1Z + ω2Z in the following way. Take a segment of a straight line through points 0 and ω of lattice L which contains no other points of L. This segment represents a homotopy class of curves through 0 and a closed geodesic of E. Evidently, this geodesic will be the shortest in its homotopy class with the length |ω| equal to an absolute value of the complex number ω.Thus,|ω| belongs to the length spectrum of E. Now let Sp E = {|ω1|, |ω2|,...} with ωi ∈ L.SinceM = αL,onegetsSp E = {|α||ω1|, |α||ω2|,...} and Sp E = |α|Sp E. Item (i) follows. (ii) Note that according to (i) the length spectrum Sp X = {l0,l1,l2,...} can be written as Sp X = {1,l1,l2,...} after multiplication on 1/l0,wherel0 is the length of the shortest geodesic. Note also that the shortest geodesic of the complex torus has a homotopy type (1, 0) or (0, 1) (standard generators for π1E). Let a, b, c, d be integers such that ad − bc = ±1andlet ω1 = aω1 + bω2, (4) ω2 = cω1 + dω2 be an automorphism of the lattice L = ω1Z + ω2Z. This automorphism maps standard generators (1, 0) and (0, 1) of L to the vectors ω1 =(a, b),ω2 =(c, d). Let their lengths be lm,lm+1, respectively. As we showed earlier, lm,lm+1 ∈ Sp E, and it is not hard to see that there are no geodesics of the intermediate length. (This gives a justification for the notation chosen.) Note that ω1,ω2 are standard generators for the complex torus E ∼ E, and therefore one of them is the shortest closed geodesics of E. One can normalize it to the length 1. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 976 IGOR NIKOLAEV On the other hand, there are only a finite number of closed geodesics of length smaller than ln (McKean [6]). Thus Sp E ∩ Sp E = {lm,lm+1,...} for a finite number m and since (4) is automorphism of the lattice L.Inotherwords,Sp E = Spm E.
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